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Van CONTRASTING INTACT AND MODEL EXPERIMENTAL A FRAMEWORK AND METHODS FOR SIMPLIFYING COMPLEX LANDSCAPES TO REDUCE UNCERTAINTY LANDSCAPE AND REGIONAL SCALE STUDIES OF MULTISCALE RELATION

Scaling and Uncertainty Analysis

in Ecology

Methods and Applications

Edited by

JIANGUO WU

Arizona State University, Tempe, AZ, U.S.A.

US Environmental Protection Agency, Las Vegas, U.S.A.

Printed on acid-free paper

All Rights Reserved

© 2006 Springer

No part of this work may be reproduced, stored in a retrieval system, or transmitted

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CONCEPTS OF SCALE AND SCALING

Jianguo Wu and Harbin Li

3

Chapter 2

PERSPECTIVES AND METHODS OF SCALING

DOWNSCALING ABUNDANCE FROM THE DISTRIBUTION

OF SPECIES: OCCUPANCY THEORY AND APPLICATIONS

Debra P C Peters, Jin Yao, Laura F Huenneke, Robert P Gibbens,

Kris M Havstad, Jeffrey E Herrick, Albert Rango, and William H

Schlesinger

131

Chapter 8

BUILDING UP WITH A TOP-DOWN APPROACH: THE ROLE

OF REMOTE SENSING IN DECIPHERING FUNCTIONAL

AND STRUCTURAL DIVERSITY

Chapter 9

CARBON FLUXES ACROSS REGIONS: OBSERVATIONAL

CONSTRAINTS AT MULTIPLE SCALES

Beverly E Law, Dave Turner, John Campbell, Michael Lefsky,

Michael Guzy, Osbert Sun, Steve Van Tuyl, and Warren Cohen

167

Chapter 10

NITROGEN GAS FLUXES

Peter M Groffman, Rodney T Venterea, Louis V Verchot, and

Christopher S Potter

191

Chapter 11

IN STREAMS

K Bruce Jones, Anne C Neale, Timothy G Wade, Chad L Cross,

James D Wickham, Maliha S Nash, Curtis M Edmonds, Kurt H

Riitters, Robert V O’Neill, Elizabeth R Smith, and Rick D Van

CONTRASTING INTACT AND MODEL EXPERIMENTAL

A FRAMEWORK AND METHODS FOR SIMPLIFYING

COMPLEX LANDSCAPES TO REDUCE UNCERTAINTY

LANDSCAPE AND REGIONAL SCALE STUDIES OF

MULTISCALE RELATIONSHIPS BETWEEN LANDSCAPE

CHARACTERISTICS AND NITROGEN CONCENTRATIONS

UNCERTAINTY IN SCALING NUTRIENT EXPORT

ASSESSING THE INFLUENCE OF SPATIAL SCALE ON THE

RELATIONSHIP BETWEEN AVIAN NESTING SUCCESS AND

FOREST FRAGMENTATION

Penn Lloyd, Thomas E Martin, Roland L Redmond, Melissa M

Hart, Ute Langner, and Ronald D Bassar

259

Chapter 15

SCALING ISSUES IN MAPPING RIPARIAN ZONES WITH

REMOTE SENSING DATA: QUANTIFYING ERRORS AND

THE NORTH CAROLINA PIEDMONT: THE SCOPE OF

SCALE ISSUES IN LAKE-WATERSHED INTERACTIONS:

ASSESSING SHORELINE DEVELOPMENT IMPACTS ON

SCALING WITH KNOWN UNCERTAINTY: A SYNTHESIS

xi

Scale is a unifying concept that cuts across all natural and social sciences At the same time, scaling is a common challenge in both basic and applied research Accordingly, scale and scaling have become two of the most widely used buzzwords

in ecology today Over the past two decades, more than a dozen books and many more journal papers have been published on the problems of scale and scaling in ecology and geophysical sciences These publications, as reviewed in the chapters of this book, have contributed significantly to our current understanding of scale issues

A little more than 30 years ago, the noted geneticist and evolutionary biologist, Theodosius Dobzhansky, stated that “Nothing in biology makes sense except in the light of evolution” (The American Biology Teacher 35:125-129) Today, there seems a growing consensus in ecology that pattern and process make little sense without consideration of scale

While scale issues are widely recognized, a comprehensive understanding of scaling theory and methods still is missing In this book we make several observations on the status of research on scale in ecology First, while ecologists have played an active role in the application of scale-related theories such as hierarchy, self-similarity, and self-organized criticality, a number of pragmatic scaling methods have developed in geophysical disciplines Many of them may be quite appropriate for a range of ecological problems, but are yet to be fully explored

in ecology Second, some of the most frequently mentioned scaling theories are often seen as being at odds with each other For example, hierarchy theory implies scale-multiplicity and thresholds, while self-similarity and self-organized criticality suggest scale invariance A full understanding of the relationships among different scaling theories is needed, and this requires critical examination of recent theoretical and empirical studies Third, most scaling studies in ecology have either ignored or inadequately addressed the issues of uncertainty and error propagation, which should be an integral part of scaling We argue that scaling, without considering uncertainty, is easy but relatively trivial; scaling with known uncertainty is challenging but essential Fourth, scaling often requires field-based data from multiple spatial and temporal scales, but these data rarely exist for many ecosystems Such inadequacies of data further elevate the demand for effective scaling

approaches Finally, scaling theories and methods have seldom been applied explicitly in the contexts of environmental management, planning, and decision-making processes, where the scale of social, economic, political, and ecological processes may clash with each other A pluralistic and interdisciplinary approach is needed to resolve scaling problems in such complex situations

To address these problems, a workshop entitled “Scaling and Uncertainty Analysis in Ecology: Methods and Application” was held during September 17-19,

2002 at Arizona State University, Tempe, U.S.A., supported through a grant from the United States Environmental Protection Agency (EPA) The major objectives of the workshop were to identify approaches and methods in scaling and uncertainty analysis, and to consider a series of case studies illustrating how scale issues are dealt with in various areas of research More than 20 active researchers in scaling and uncertainty analysis were invited to participate in the workshop, many of whom were recipients of EPA’s Science To Achieve Results (STAR) program (Regional Scale Analysis and Assessment) This book has evolved out of the scaling workshop, and is comprised primarily of the papers remaining after a critical external review process

The book, therefore, presents a comprehensive and up-to-date review and synthesis of concepts, theories, methods and case studies in scaling and uncertainty analysis that are relevant to ecology The series of case studies included here illustrate how scaling and uncertainty analysis are being conducted in ecology and environmental science, from population to ecosystem processes, from biodiversity to landscape patterns, and from basic research to multidisciplinary management and policy-making issues The book explicitly considers uncertainty and error analysis

as an integral part of scaling While the theme of this book focuses primarily on spatial scaling, several chapters deal as well with aspects of temporal scaling It is not intended to be a handbook of “scaling recipes,” but we hope that it will help readers gain a fuller understanding of the state-of-the-science of scale issues We expect that this book will be of interest to a wide range of audiences, including graduate students, academic professionals, and applied researchers and specialists in ecological, environmental, and earth sciences It may be used as a text or reference book for graduate courses in ecology and related disciplines This book should be particularly appealing to scientists and practitioners working on broad spatial scales Also, the book can be useful to decision makers who are conscious about scale issues as they translate science into resource use policies

We are most deeply indebted to the contributors of papers included in the book, whose enthusiasm and dedication have made this book a reality Many other individuals also were instrumental to the completion of the book We especially thank the following people for providing valuable reviews of book chapters: Dennis Baldocchi, Klaus Butterbach-Bahl, Mark Castro, Jiquan Chen, Mark R T Dale, Dean Gesch, Phil A Graniero, John Harte, Geoffrey J Hay, Louis R Iverson, James

R Karr, Madhu Katti, Richard G Lathrop, Helene Muller-Landau, John Ludwig, James R Meadowcroft, Garry Peterson, Geoffrey C Poole, Edward B Rastetter, Helen Regan, Christine Ribic, Steven W Running, Santiago Saura, Matthew Williams, and Xinyuan Wu We are extremely grateful to Chuck Redman (Director), Nikol Grant, and Shirley Stapleton at the Center for Environmental Studies of Arizona State University who provided wonderful logistic support during the scaling workshop in Tempe We also thank Barbara Levinson and Jonathan Smith at EPA

for their support for the scaling workshop in Tempe Last, but not least, we express our sincere appreciation to Dr Catherine Cotton (Publishing Editor) and Ms Ria Kanters at Springer for their wonderful guidance and assistance during the production of the book

Finally, we should note that several chapters originally had color images which later were converted to grayscale We have made these color figures available online

at a web site specifically for this book, which also contains the abstracts of all chapters and additional information on scaling and uncertainty analysis The web address can be freely accessed at: http://LEML.asu.edu/ScalingBook/

Editors Jianguo (Jingle) Wu

K Bruce Jones Harbin Li Orie L Loucks

Curtis M Edmonds, U.S Environmental Protection Agency, Las Vegas, NV

Kris M Havstad, USDA ARS, Jornada Experimental Range, Las Cruces, NM 88003-0003

Fangliang He, Department of Renewable Resources, University of Alberta, Edmonton, Alberta, Canada T6G 2H1

Jeffrey E Herrick, USDA ARS, Jornada Experimental Range, Las Cruces, NM 88003-0003

Thomas P Hollenhorst, Natural Resources Research Institute, University of Minnesota, Duluth, MN 55811-1442

George E Host, Natural Resources Research Institute, University of Minnesota, Duluth, MN 55811-1442

Laura F Huenneke, College of Engineering and Natural Sciences, Northern Arizona University, Flagstaff, AZ 86011

K Bruce Jones, U.S Environmental Protection Agency, Las Vegas, NV 89193-3478 Lucinda B Johnson, Natural Resources Research Institute, University of Minnesota, Duluth, MN 55811-1442

Carol A Johnston, Center for Biocomplexity Studies, South Dakota State University, Brookings, SD 57007-0896

Bruce M Kahn, Department of Zoology, Miami University, Oxford, OH 45056 Ute Langner, Montana Cooperative Wildlife Research Unit, University of Montana, Missoula, MT 59812

Beverly E Law, College of Forestry, Oregon State University, Corvallis, OR

97331-5752

Michael Lefsky, College of Forestry, Oregon State University, Corvallis, OR 97331 Harbin Li, USDA Forest Service Southern Research Station, Center for Forested Wetlands Research, Charleston, SC 29414

Penn Lloyd, DST/NRF Centre of Excellence, Percy FitzPatrick Institute, University

of Cape Town, Rondebosch, 7701, South Africa

Orie L Loucks, Department of Zoology, Miami University, Oxford, OH 45056 Thomas E Martin, Montana Cooperative Wildlife Research Unit, University of Montana, Missoula, MT 59812

Robert I McDonald, Nicholas School of Environment & Earth Sciences, Duke University, Durham, NC 27708

Emily S Minor, Nicholas School of Environment & Earth Sciences, Duke University, Durham, NC 27708

Maliha S Nash, U.S Environmental Protection Agency, Las Vegas, NV

89193-3478

Anne C Neale, U.S Environmental Protection Agency, Las Vegas, NV 89193-3478 Robert V O’Neill, Environmental Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831

Debra P.C Peters, USDA ARS, Jornada Experimental Range, Las Cruces, NM 88003-0003

Christopher S Potter, NASA Ames Research Center, Moffett Field, CA 94035-1000 Albert Rango, USDA ARS, Jornada Experimental Range, Las Cruces, NM 88003-

Rick D Van Remortel, Lockheed Martin Environmental Services, Las Vegas, NV

89119

Steve Van Tuyl, College of Forestry, Oregon State University, Corvallis, OR 97331 Rodney T Venterea, Institute of Ecosystem Studies, Box AB, Millbrook, NY 12545 Louis V Verchot, Institute of Ecosystem Studies, Box AB, Millbrook, NY 12545

Timothy G Wade, U.S Environmental Protection Agency (E243-05), National Exposure Research Laboratory, Research Triangle Park, NC 27711

Carol A Wessman, Cooperative Institute for Research in Environmental Sciences and Department of Ecology and Evolutionary Biology, University of Colorado, Boulder, CO 80309-0216

James D Wickham, U.S Environmental Protection Agency (E243-05), National Exposure Research Laboratory, Research Triangle Park, NC 27711

Jianguo (Jingle) Wu, School of Life Sciences and Global Institute of Sustainability, Arizona State University, Tempe, AZ 85287-4501

Jin Yao, Department of Biology and Earth Sciences, Adams State College, Alamosa,

CO 81102

PART I CONCEPTS AND METHODS

3

J Wu, K.B Jones, H Li, and O.L Loucks (eds.),

Scaling and Uncertainty Analysis in Ecology: Methods and Applications, 3–15

© 2006 Springer Printed in the Netherlands

CONCEPTS OF SCALE AND SCALING

JIANGUO WU AND HARBIN LI

1.1 INTRODUCTION The relationship between pattern and process is of great interest in all natural and social sciences, and scale is an integral part of this relationship It is now well documented that biophysical and socioeconomic patterns and processes operate on a wide range of spatial and temporal scales In particular, the scale multiplicity and scale dependence of pattern, process, and their relationships have become a central topic in ecology (Levin 1992, Wu and Loucks 1995, Peterson and Parker 1998) Perspectives centering on scale and scaling began to surge in the mid-1980’s and are pervasive in all areas of ecology today (Figure 1.1) A similar trend of increasing emphasis on scale and scaling is also evident in other natural and social sciences (e.g., Blöschl and Sivapalan 1995, Marceau 1999, Meadowcroft 2002)

Scale usually refers to the spatial or temporal dimension of a phenomenon, and

scaling is the transfer of information between scales (more detail below) Three distinctive but interrelated issues of scale have frequently been discussed in the literature: (1) characteristic scales, (2) scale effects, and (3) scaling (and associated

uncertainty analysis and accuracy assessment) The concept of characteristic scale

implies that many, if not most, natural phenomena have their own distinctive scales (or ranges of scales) that characterize their behavior (e.g., typical spatial extent or event frequency) Characteristic scales are intrinsic to the phenomena of concern, but detected characteristic scales with the involvement of the observer may be tinted with subjectivity (Wu 1999) Conceptually, characteristic scales may be perceived as

the levels in a hierarchy, and associated with scale breaks (O’Neill et al 1991, Wu

1999) Ecological patterns and processes have been shown to have distinctive characteristic scales on which their dynamics can be most effectively studied (Clark

1985, Delcourt and Delcourt 1988, Wu 1999) Thus, identifying characteristic scales provides a key to profound understanding and enlightened scaling

Scale effects usually refer to the changes in the result of a study due to a change

in the scale at which the study is conducted Effects of changing scale on sampling

and experimental design, statistical analyses, and modeling have been well documented in ecology and geography (e.g., Turner et al 1989b, White and Running 1994, Wu and Levin 1994, Pierce and Running 1995, Jelinski and Wu

1996, Dungan et al 2002, Wu 2004) In geography, scale effects have been studied

for several decades in the context of the modifiable areal unit problem or MAUP

(Openshaw 1984, Jelinski and Wu 1996, Marceau 1999) Scale effects may be explained in terms of scale-multiplicity, characteristic scales, and hierarchy, but may also be artifacts due to errors in sampling and measurements, distortions in data resampling, and flaws in statistical analysis and modeling (Jelinski and Wu 1996,

Wu 2004) Characteristic scales and scale effects are inherently related to the issue

of scaling While characteristic scales provide a conceptual basis and practical guidelines for scaling, quantitative descriptions of scale effects can directly lead to scaling relations (Wu 2004)

Figure 1.1 Rapid increase in the use of terms related to scale in the ecological literature

Based on an internet search using JSTOR (http://www.jstor.org/), the number of articles containing words (scaling, hierarchy, hierarchies, hierarchical, hierarchy theory) shows a great increase in four major ecology journals in the last seven decades (gray line) The trend for the word scaling alone is similar (black line) The four journals are: Ecology and Ecological Monographs published by Ecological Society of America, and Journal of Ecology and Journal of Animal Ecology published by British Ecological Society Note that the number

of years for the 1990’s was only seven (1990-1996)

With the recent burst of interest in the issues of scale, the terms scale and scaling

have become buzzwords in ecology However, because these terms have been used

in diverse disciplines, both have acquired a number of different connotations and expressions Good science starts with clear definitions The development of a science of scale or scaling may be hampered if the concepts of scale and scaling are

used without any consistency In this section, we review the main usages of these

terms, propose a three-tiered scale conceptualization framework, and discuss their

relevance to the issue of ecological scaling

1.2 CONCEPT OF SCALE

We propose a three-tiered conceptualization of scale, which organizes scale

definitions into a conceptual hierarchy that consists of the dimensions, kinds, and

components of scale (Figure 1.2) Dimensions of scale are most general, components

of scale are most specific, and kinds of scale are in between This three-tiered

structure seems to provide a clearer picture of how various scale concepts differ

from or relate to each other

1.2.1 Dimensions of Scale

We distinguish three primary dimensions of scale: space, time, and organizational

level Note that Dungan et al.’s (2002) three dimensions of scale (sampling, analysis,

and phenomena) are commensurable with what we here call the kinds of scale (see

below) Space and time are the two fundamental axes of scale, whereas organizational

hierarchies are usually constructed by the observer (Figure 1.2a) Scale has been

commonly defined in terms of time or space In recent decades, the relationship

between temporal and spatial scales has received increasing attention It is well

documented that the characteristic scales of many physical and ecological phenomena

are related in space versus time, such that the ratio between spatial and temporal

scales tends to be relatively invariant over a range of scales This ratio is termed the

For the purpose of scaling, levels of organization or integration are most useful

when they are consistent with spatial and temporal scales Hierarchy theory states

that higher levels are larger and slower than lower levels, which is consistent with

the space-time principle This is generally true for nested hierarchies (i.e., systems

characteristic velocity (Blöschl and Sivapalan 1995) The idea that spatial and

temporal scales are fundamentally linked so that complex systems can be decomposed

in time and space simultaneously is essential to hierarchy theory (Courtois 1985, Wu

1999) This space-time correspondence principle has been supported by a number of

empirically constructed space-time scale diagrams (or Stommel diagrams) in the past

two decades (Stommel 1963, Clark 1985, Urban et al 1987, Delcourt and Delcourt

1988, Blöschl and Sivapalan 1995, Wu 1999) These studies have shown that, for a

variety of physical, ecological, and socioeconomic phenomena, large-sized events

tend to have slower rates and lower frequencies, whereas small things are faster and

more frequent However, one must recognize that not all natural phenomena strictly

obey the space-time correspondence principle Many temporally cyclic events, for

example, take place over a wide range of spatial scales with a relatively constant

frequency In some other cases, scale variability of different sources may overwhelm

the signal of scale correspondence Furthermore, the space-time scale ratio of most

ecological phenomena can surely be altered drastically by human modifications

in which small entities are contained by larger entities which are in turn contained

by even larger entities), but not for non-nested hierarchies (Wu 1999) In this view, the three dimensions of scale – space, time and organizational or integrative levels – can be related to each other When moving up the ladder of hierarchical levels, the characteristic scales of entities or events in both space and time also tend to change accordingly

Figure 1.2 A hierarchy of scale concepts: (A) dimensions of scale, (B) kinds of scale, and (C)

components of scale (A was modified from Dungan et al 2002; B and C were based on Bierkens et al 2000)

1.2.2 Kinds of Scale

Several kinds of scale can be distinguished based on any of the three dimensions of

scale (Figure 1.2b) Intrinsic scale refers to the scale on which a pattern or process

actually operates, which is similar to, but broader than, the concept of process scale,

a term frequently used in earth sciences (e.g., Blöschl and Sivapalan 1995) Some

may argue that there is no intrinsic scale in nature, and that scales or hierarchical

levels are merely epistemological consequences of the observer (Allen and Starr

1992) We believe that the observed scale of a given phenomenon is the result of the

interaction between the observer and the inherent scale of the phenomenon

Although the existence of intrinsic scales does not mean that they are always readily

observable, a suite of methods, including spectral analysis, fractal analysis, wavelet

analysis, scale variance, geostatistics, and multiscale object-specific analysis (e.g.,

Turner et al 1991, Wu et al 2000, Hay et al 2001, Dale et al 2002, Hall et al

2004), have been used in detecting characteristic scales or scale breaks Effective

scale detection requires that the scale of analysis be commensurate with the intrinsic

scale of the phenomenon under study (Blöschl and Sivapalan 1995, Wu and Loucks

1995, Dungan et al 2002, Legendre et al 2002) Because the latter is unknown a

priori, multiple observation sets at different scales usually are necessary (Allen et al

1984, Wu 1999)

There are several other kinds of scale that are not intrinsic to the phenomenon of

interest Observational scale is the scale at which sampling or measurement is taken

(also referred to as sampling scale or measurement scale) In experimentation, the

spatial and temporal dimensions of the experimental system represent the

experimental scale, which is the primary criterion for distinguishing among micro-,

meso-, and macro-scale experiments Similarly, the resolution and extent in space

and time of statistical analyses and dynamic models define the analysis scale or

modeling scale In the context of environmental management and planning, local,

regional, and national laws and regulations introduce another kind of scale – the

policy scale, which is influenced by a suite of economic, political, and social factors

These different kinds of scales are related to each other in various ways (Figure

1.2b) In general, only when the scales of observation and analysis are properly

chosen, may the characteristic scale of the phenomenon of interest be detected

correctly; only when the scales of experiments and models are appropriate, may the

results of experiments and models be relevant; only when the scale of

implementation of policies is commensurate with the intrinsic scale of the problem

under consideration, may the policies be effective In reality, different kinds of

scales may differ even for the same phenomenon, resulting in the problem of scale

mismatch (or scale discordance) To rectify such scale mismatch or to relate one

type of scale to the other usually involves scale transfer or scaling (Bierkens et al

2000) An adequate understanding of the relationship among the different kinds of

scale needs to invoke the definitions of scale components

1.2.3 Components of Scale

Dimensions of scale and kinds of scale are useful general concepts, but more specific and measurable definitions are required in order to quantify scale and

develop scaling relations These are the components of scale, including cartographic

scale, grain, extent, coverage, and spacing (Figure 1.2c) The traditional

cartographic scale (or map scale) is the ratio of map distance to actual distance on

the earth surface A so-called large-scale map usually covers a smaller area with greater detail Cartographic scale is essential for the creation and use of maps, but inadequate for studying the scale-dependent relationships between pattern and process in ecology because of its intended rigid connotation (Jenerette and Wu 2000)

In ecology and other earth sciences, scale most frequently refers to grain and

extent – two primary components of scale Grain is the finest resolution of a

phenomenon or a data set in space or time within which homogeneity is assumed, whereas extent is the total spatial or temporal expanse of a study (Turner et al 1989a, Wiens 1989) Grain may be considered as the pixel size for raster data, or the minimum mapping unit for vector data A frequently used geostatistical term,

support, refers to the smallest area or volume over which the average value of a

variable is derived (Dungan et al 2002) In most cases, grain and support have quite similar meanings, and thus have often been used interchangeably However, support may differ from grain because support itself includes not only the size of an n-dimensional volume, but also its geometrical shape, size and orientation (Dungan

et al 2002) When the linear or areal dimension of grain is referred to, grain element

or grain unit can be used, which corresponds to support unit in the literature Note

that soil scientists and hydrologists frequently use scale only to refer to support (e.g., Bierkens et al 2000)

On the other hand, the concept of extent is less diversified than grain A term equivalent to extent is geographic scale, which was defined by Lam and Quattrochi

(1992) as the size of a particular map Both grain and extent are of great importance

to the study of heterogeneous landscapes (Turner 1989) Besides grain and extent, coverage and spacing, which are associated particularly with sampling, are also

important in scaling Coverage, not to be confused with extent, refers to sampling intensity in space or time (Bierkens et al 2000), while spacing is the interval

between two adjacent samples or lag Spatial coverage can be represented as the ratio of the sampled area to the extent of a study, and spacing may be fixed or variable depending on the sampling scheme (Figure 1.2c) Support, extent, and

spacing are sometimes called the scale triplet in hydrological literature, which

highlights the importance of these three components in scaling (Blöschl and Sivapalan 1995)

The relationship between intrinsic scale and other kinds of scales can be further elaborated in terms of scale components Hierarchy theory suggests that the scale of observation must be commensurate with the scale of the phenomenon under consideration if the phenomenon is to be properly observed (Simon 1973, Allen

et al 1984, O’Neill et al 1986, Wu 1999) On the one hand, processes larger than the extent of observation appear as trends or constants in the observation set; on the

other hand, processes smaller than the grain size of observation become noise in the

data Thus, the choice of a particular scale for observation, analysis and modeling in

terms of grain size and extent directly influences whether or not the intrinsic pattern

and scale of a phenomenon can be eventually revealed in the final analysis The

significance of the choice of scale has long been recognized in plant ecology (e.g

Greig-Smith 1983) and human geography (Openshaw 1984, Jelinski and Wu 1996)

In general, the grain size of sampling or observation should be smaller than the

spatial or temporal dimension of the structures or patterns of interest, whereas it is

desirable to have the sampling extent at least as large as the extent of the

phenomenon under study (Dungan et al 2002)

In addition, the concept of relative scale can be rather useful for comparative

studies and scaling across different ecosystems or landscapes Meentemeyer (1989)

defined relative scale as the relationship between the smallest distinguishable unit

and the extent of the map, which can be expressed simply as a ratio between grain

and extent Schneider (2001) used range to refer to extent, and defined scope as the

ratio of the range to the resolution of a research design, a model, or a process In

principle, different phenomena and research designs can be compared on the basis of

their scopes Relative scale can also be defined by directly incorporating the

ecological pattern and process under consideration Such definition is rooted in the

conceptualization of relative versus absolute space (Meentemeyer 1989, Marceau

1.3 CONCEPT OF SCALING Scaling has been defined differently in various fields of study, and its meanings can

be quite disparate Scaling has long been associated with measurement that is “the

assignment of numerals to objects or events according to rules” (Stevens 1946) In

this case, scaling is a way of measuring the “unmeasurable” (Torgerson 1958) In

multivariate statistics, scaling usually refers to a set of techniques for data reduction

and detection of underlying relationships between variables Multivariate statistical

methods, such as polar ordination, multidimensional scaling, principal component

analysis, and correspondence analysis, have been used extensively in vegetation

classification and ordination to organize field plots (or community types) into some

order according to their similarities (or dissimilarities) on the basis of species

composition Multidimensional scaling, in particular, is used to represent similarities

among objects of interest through visual representation of Euclidean space-based

patterns, and has been widely used to analyze subjective evaluations of pairwise

similarities of entities in a wide range of fields, including psychology, marketing,

sociology, political science, and biology (Young and Hamer 1994) These

multivariate statistical methods can be useful for relating patterns and processes

across scales (e.g., multiscale ordination; ver Hoef and Glenn-Lewin 1989)

However, the concept of scaling as either the assignment of numerical values to

1999) For example, Turner et al (1989b) considered relative scale as “a transformation

of absolute scale to a scale that describes the relative distance, direction, or

geometry based on some functional relationship (e.g., the relative distance between

two locations based on the effort required by an organism to move between them).”

qualitative variables or the reduction and ordination of data is not directly relevant to scaling as defined below

In physical sciences, scaling usually refers to the study of how the structure and

behavior of a system vary with its size, and this often amounts to the derivation of a

power-law relationship This notion of scaling has often been related to the concepts

of similarity, fractals, or scale-invariance, all of which are associated with power laws For example, a phenomenon or process is said to exhibit “scaling” if it does not have any characteristic length scale; that is, its behavior is independent of scale – i.e., a power law relationship (Wood 1998) This definition of scaling has long been

adopted by biologists in terms of allometry that primarily correlates the size of

organisms with biological form and process (Wu and Li, Chapter 2) In this context, scale refers to “the proportion that a representation of an object or system bears to the prototype of the object or system” (Niklas 1994), and ecological scaling then becomes “the study of the influence of body size on form and function” (LaBarbera

1989) Thus, to some, ecological scaling is simply some form of biological

allometry (e.g., Calder 1983, Schmidt-Nielsen 1984, LaBarbera 1989, Brown and West 2000)

Several other terms are closely related to, but not the same as, scaling These terms are associated with three basic scaling operations: changing extent, changing

grain size, and changing coverage Extrapolation is transferring information from smaller to larger extents, coarse-graining transferring information with increasing grain size, and fine-graining transferring information with decreasing grain size

Sometimes, upscaling and downscaling refer specifically to coarse-graining and fine-graining, respectively (e.g., Bierkens et al 2000) When dealing with spatial data that do not have 100% coverage, one may need to estimate the values of unmeasured spatial locations using information from measured sites – a process

called interpolation The reverse process of interpolation is sampling In practice,

the three basic operations may all be needed in a single study That is, different

However, a more general and widely accepted definition of scaling in ecology

and earth sciences is the translation of information between or across spatial and

information can be done through explicit mathematical expressions and statistical relationships (scaling equations), whereas in many other cases process-based mulation models are necessary This definition of scaling is also referred to as

scale transfer or scale transformation (Blöschl and Sivapalan 1995, Bierkens et al

2000) This broadly defined scaling concept neither implies that scaling relations must be power-laws, nor that ecological patterns and processes must show scale-independent properties in order to “scale” or to be “scaled.” In this case, allometric scaling is but only one special case of scaling Based on the directionality of the

scaling operation, two kinds of scaling can be further distinguished: (1) scaling up or

upscaling which is translating information from finer scales (smaller grain sizes or

extents) to broader scales (large grain sizes and extents), and (2) scaling down or

downscaling which is translating information from broader scales to finer scales

al 2000, Gardner et al 2001) In some cases, this across-scale translation of Sivapalan 1995, Stewart et al 1996, van Gardingen et al 1998, Wu 1999, Bierkensemporal scales or organizational levels (Turner et al 1989a, King 1991, Blöschl and et

methods for interpolation, sampling, coarse-graining, fine-graining, and extrapolation

may be used together to achieve the overall goal of scaling In general, to make the

concept of scale operational, one needs to be specific about the scale components (e.g.,

grain, extent, coverage, spacing) To put the concept of scaling into action, one has to

invoke specific scaling operations (e.g., extrapolation, coarse-graining, fine-graining,

interpolation) Any spatial scaling approach or method will inevitably involve one or

more of the basic scaling operations

Note that the definition of extrapolation given above is quite specific and

unequivocal However, in the literature, extrapolation in space has been used in at

least four distinct ways: (1) using known data acquired from certain locations to

estimate unknown values or draw inferences at other locations, (2) estimating values

or drawing inferences about things that fall outside the study area, (3) transferring

information from one scale to another in terms of either extent or grain, and (4)

transferring information between different systems at the same spatial scale (Turner

1.4 WHY SCALING AND HOW?

Simply put, scaling is the essence of prediction and understanding, and is at the heart

of ecological theory and application (Levin 1992, Levin and Pacala 1997, Wu 1999,

Chave and Levin 2003) More specifically, two main reasons are commonly

recognized First, scaling is inevitable in research and practice whenever predictions

need to be made at a scale that is different from the scale where data are acquired In

general, whenever information is averaged over space or time, scaling is at work

For example, the sampling plots that ecologists usually use for determining the

distribution of organisms or the stocks and fluxes of materials are only a small

portion of the spatial extent of ecological systems of interest Thus, system-level

descriptions dictate the translation of information from these small plots to much

larger areas Also, while most ecological studies traditionally have been conducted

on local scales, environmental and resource management problems often have to be

dealt with on much broader scales (i.e., landscapes, regions, or the entire globe) To

bridge such scale gaps requires scaling

Second, because ecological phenomena occur over a wide range of scales and

because there are often hierarchical linkages among them, relating information

across scales as well as levels of biological organization is an essential part of

ecological understanding For example, the dynamics of sub-watershed units and

their interactions are crucial to understanding the hydrological and biogeochemical

cycles of the whole watershed ecosystem (Wickham et al., Chapter 12) The

dynamics of local populations and their interpatch interactions are crucial to

et al 1989a, Wu 1999) The multiple meanings of extrapolation may cause confusions

For example, the first usage is simply spatial interpolation The second is consistent

with the definition of spatial extrapolation as information transfer with increasing

extent The third is extremely broad and may refer to coarse-graining, fine-graining,

or scaling in general The fourth usage makes sense with regard to the literal

meaning of the word, but it does not fit the definition of scaling because scaling has

to involve at least two or more scales Hence, the term extrapolation should be used

with caution

understanding population dynamics at the landscape scale In a similar vein, understanding the primary productivity of the whole ecosystem requires knowledge

of photosynthesis at the individual leaf level

While it is imperative in almost all ecological studies, spatial scaling can also be extremely challenging in theory and practice Spatial heterogeneity can greatly complicate the scaling process Spatial heterogeneity may manifest itself in terms of various patterns of land use and land cover, topography, hydrology, soils, climatic conditions, and biological factors For example, extrapolation of plot-scale data to the landscape or regional scale is a trivial matter in a spatially homogeneous (uniform or random) environment In a heterogeneous landscape, however, simply multiplying the plot-scale average with the total study area usually provides a rather poor estimate at the landscape scale (Li and Wu, Chapter 3) When ecological relationships are translated across scales in heterogeneous environments, they often

become distorted – a phenomenon known as “spatial transmutation” (sensu O’Neill

1979, King et al 1991, Wu and Levin 1994)

Also, as scale changes, new patterns and processes may emerge, and controlling factors may shift even for the same phenomena Thus, observations made at fine scales may miss important patterns and processes operating on broader scales Conversely, broad-scale observations may not have enough details necessary to understand fine-scale dynamics In addition, nonlinear interactions, time delays, feedbacks, and legacies in ecological systems may impose formidable challenges for translating information across scales or levels of organization (O’Neill and Rust 1979, Wu 1999) Therefore, on the one hand, spatial heterogeneity, scale multiplicity, and nonlinearity are important sources of biodiversity and ecological complexity; on the other hand, they are major hurdles for successful scaling

Given the various obstacles, how should we proceed with scaling? This is the focus of our next chapter, where we will discuss two general scaling approaches: similarity-based and dynamic model-based scaling A dozen specific scaling methods will also be examined in terms of their assumptions, ways of dealing with spatial heterogeneity and nonlinear interactions, and accuracy of scaling results No matter which approach is used, an important concept in scaling up and down is

scaling threshold or scaling break, which signifies a narrow range of scale around

which scaling relations change abruptly A scaling threshold may also be understood

as a critical scale of a phenomenon where emergent properties due to nonlinear interactions and spatial heterogeneity come into effect Thus, scaling thresholds, when properly identified, may reflect fundamental shifts in underlying processes or controlling factors, and can be used to define the domains of applicability of specific scaling methods

1.5 DISCUSSION

In this chapter, we have discussed and clarified a number of concepts related to scale and scaling as used in a variety of fields of study We propose a hierarchical framework in which the different connotations of scale can be organized with clarity and consistency The three-tiered definitional hierarchy, consisting of the dimensions, kinds, and components of scale, shows both the diversity and interrelatedness of the

concepts of scale In the practice of scaling, the components of scale (most

frequently extent, grain, and coverage) must be invoked Indeed, scaling methods

are often designed to capture and deal with the change in these scale components

singularly or in concert (see Wu and Li, Chapter 2 for details)

Clarification of key concepts is the first step towards a science of scale The

three-tiered definitional hierarchy seems to serve this purpose well even though it is

only one of many possible ways of organizing these concepts It is crucial for

ecologists to recognize the different usages of scale and scaling, and to adopt a

system of definitions that are consistent, clear, and accommodating to the

development of quantitative methods The science of scale will certainly benefit

from clear concepts and definitions, which are essential for the development of

effective methods and sound theories of scaling

ACKNOWLEDGEMENTS

We would like to thank Geoffrey Hay, Fangliang He, Bruce Jones, and Simon Levin

for their comments on an earlier version of the chapter JW’s research on scaling has

been supported in part by grants from U.S Environmental Protection Agency’s

Science to Achieve Results (STAR) Program (R827676-01-0) and National Science

Foundation (DEB 9714833 for Central Arizona-Phoenix Long-Term Ecological

Research)

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© 2006 Springer Printed in the Netherlands

PERSPECTIVES AND METHODS OF SCALING

JIANGUO WU AND HARBIN LI

2.1 INTRODUCTION Transferring information between scales or organizational levels is generally

referred to as “scaling” (Wu and Li, Chapter 1), and is inevitable in both basic

research and its applications Scaling is the essence of prediction and understanding both of which require cross-scale translation of information, and is at the core of ecological theory and application (Levin 1992, Levin and Pacala 1997, Wu 1999) While the importance of scaling in ecology has been acutely recognized in recent decades, how to conduct scaling across heterogeneous ecosystems remains a grand challenge (Turner et al 1989, Wu and Hobbs 2002)

A number of scaling approaches and methods have been developed and applied

in different disciplines ranging from physics, engineering, biology, to social sciences Two general scaling approaches can be distinguished: similarity-based scaling and dynamic model-based scaling methods (Blöschl and Sivapalan 1995)

Similarity-based scaling methods are rooted in the concepts and principles of

While the previous chapter (Wu and Li, Chapter 1) discussed various concepts of scale and scaling, in this chapter we focus on the major characteristics of the two scaling approaches and several more specific upscaling and downscaling methods

similarity and self-similarity and often characterized by relatively simple mathematical

or statistical scaling functions, even though the underlying ecological processes of a

phenomenon may be extremely complex In contrast, dynamic model-based scaling

methods use deterministic or stochastic models to simulate the processes of interest,

and to transfer information across scales by either modifying the parameters and input variables of the same model or developing multiple-scaled models In this case, information transfer between different scales is accomplished through manipulating the inputs, outputs, and formulations of dynamic models In both approaches, it is important to properly identify scaling thresholds at which scaling relations often change abruptly, reflecting fundamental shifts in underlying processes

or controlling factors and defining the domains of applicability of specific scaling methods

within each approach The purpose of this chapter is not to provide a recipe for scaling Rather, we shall review scaling perspectives and methods in different disciplines, and provide a synthesis based on a common conceptual framework By

so doing, we expect that a more comprehensive and cohesive understanding of ecological scaling will emerge

2.2 SIMILARITY-BASED SCALING METHODS

2.2.1 The Concept of Similarity

The concept of similarity has been essential in scaling-related studies In general,

similarity exists between two systems whenever they share some properties that can

be related across the systems by a simple conversion factor (Blöschl and Sivapalan 1995) LaBarbera (1989) summarized three types of similarity concepts applied in body size-oriented studies: geometric, physical, and functional similarities (also see

Gunther 1975) Geometric similarity is characterized by the constancy in shape with

changing size In other words, geometric similarity assumes that “geometry and shape are size-independent properties” (Niklas 1994) For example, for different sized objects of the same shape and geometry, S ∝ L2, and S ∝ V2 / 3, where L, S, and

V are the linear dimension, surface area, and volume of the objects Physical similarity is defined based on the constancy of the ratios of different forces (also

called dynamic similarity; see Blöschl and Sivapalan 1995) For example, two systems are said to have hydrodynamic similarity if they have the same Reynolds

number (i.e., the ratio of inertial to viscous forces) Barenblatt (1996) stated that the concept of physical similarity is a natural generalization of that of geometric similarity in that two similar triangles differ only in the numerical values of side lengths, whereas two similar physical phenomena differ only in the numerical values

of the dimensional governing parameters Functional similarity refers to the

constancy in changes of functional variables over a range of system sizes For

example, animal metabolic rates (R) change with body size or mass (M ) following a power law (i.e., R ∝ M b ) Similarly, if the primary productivity (P) of a group of ecosystems changes with their spatial extent (A) in a power-law fashion (i.e.,

P ∝ A b), then these ecosystems may be said to have functional similarity

In recent decades, the concept of self-similarity has become a cornerstone of

similarity-based studies It refers to the phenomenon that the whole is composed of smaller parts that resemble the whole itself or that patterns remain similar at different scales Self-similarity is the key idea in fractal geometry (Mandelbrot 1982, Hastings and Sugihara 1993), and is considered to be the unifying concept underlying fractals, chaos, and power laws (Schroeder 1991) While admitting that the terms, fractal and multifractal, still lack an agreed mathematical definition,

Mandelbrot (1999) offered an informal definition of fractal geometry as “the

systematic study of certain very irregular shapes, in either mathematics or nature, wherein each small part is very much like a reduced size image of the whole.” Such irregular shapes, or fractals, exhibit properties of self-similarity which entails scale-invariance (i.e., patterns or relationships remain unchanged over a range of scales)

Commonly cited examples of fractals include coastlines, clouds, snowflakes,

branching trees, and vegetation patches However, not all self-similar objects are

fractals because self-similarity is also found in Euclidean geometry

Simple fractals exhibit scale-invariant patterns that can be characterized with

only one scaling exponent, which is often interpreted as implying one single

generating process However, many fractal-like structures in nature are generated by

a number of generating processes that operate at different scales These are called

generalized fractals, or multifractals, which are characterized by a spectrum of

fractal dimensions that vary with scale It has been suggested that additive processes

tend to create monofractals (simple fractals), whereas multiplicative random

processes generate multifractal structures (Stanley and Meakin 1988, Schroeder

1991) Multifractals have been used to describe the spatial distribution of people and

minerals, energy dissipation in turbulence, and many other patterns and processes in

nature It is now widely recognized that many, if not most, fractal patterns and

processes in nature show scale-invariance only over a limited range of scales

Hastings and Sugihara (1993) suggested that linear regression methods be used to

distinguish between patterns with one scaling region (a single power law) and those

with multiple scaling regions (separate power laws over separate regions) These

authors asserted that multiscaling is detected if the slope of the regression line

changes significantly over adjacent regions

2.2.2 Dimensional Analysis and Similarity Analysis

The concepts of similarity are the foundation of dimensional analysis (Blöschl and

Sivapalan 1995), and have long been used in engineering and physical sciences

Barenblatt (1996) indicated that the main idea behind dimensional analysis is that

“physical laws do not depend on arbitrarily chosen basic units of measurement,” and

thus the functions expressing physical laws must possess some fundamental property

(mathematically termed generalized homogeneity or symmetry) that allows the

number of arguments in these functions to be reduced Dimensional analysis aims to

produce dimensionless ratio-based equations that can be applied at different scales

for a phenomenon under study In practice, dimensional analysis only applies in the

framework of Euclidean geometry and Newtonian dynamics (Scheurer et al 2001)

Examples of similarity analysis are abundant in physical sciences For example,

similarity analysis in soil physics and hydrology started in the 1950s with the

concept of Miller-Miller similitude, an intuitive depiction of structural similarities in

porous media at fine spatial scales (Miller and Miller 1956) Miller-similar porous

media have microscopic structures that look similar in the same way as triangles in

Dimensional techniques have long been used to derive similarity relationships,

establish scaling laws, reduce data volume, and help elucidate processes and

mechanisms in physical and biological sciences (Gunther 1975, Blöschl and Sivapalan

1995) Like dimensional analysis, similarity analysis is also a simplification

procedure to replace dimensional quantities required for describing a phenomenon

with fewer dimensionless quantities; but unlike dimensional analysis, similarity

analysis requires the governing equations of the phenomenon to be known (Blöschl

and Sivapalan 1995)

Euclidean geometry (Sposito 1998) Similarity analysis, then, involves the derivation of scale factors for soil-water transport coefficients on the basis of the fine-scale similar-media concept Later studies extended the concept of the Miller-Miller similitude from microscopic to macroscopic scales using the idea of functional normalization (related to functional similarity) rather than dimensional techniques (Haverkamp et al 1998) In recent decades, fractal and multifractal models of soil structure have been increasingly used in similarity analysis of hydrological processes and beyond As Sposito (1998) noted, “fractal geometry has become the signature approach to both spatial-scale invariance and temporal-scale invariance, as epitomized by self-similarity in the patterns of hydrologic and other geophysical processes.”

One of the most successful examples of using similarity analysis to deal with complex physical processes is Monin-Obukhov similarity theory Atmospheric

boundary-layer flows, though mostly turbulent, can be viewed as being dynamically similar such that the concepts of similarity can provide a powerful framework for analyzing empirical data and parameterizing models to represent these complex processes In particular, Monin-Obukhov theory assumes that surface layers with the same ratio of the aerodynamic roughness length (z 0) to the Obukhov length (L) are

dynamically similar, with z 0 /L being considered as a dimensionless similarity

parameter In other words, the theory is based on the assumption of complete

As an important part of similarity analysis, renormalization group methods

(Wilson 1975) have been used for studying scaling behavior associated with critical phenomena and phase transitions in physical sciences, including turbulence, flows in porous media, fracture mechanics, flame propagation, atmospheric and oceanic processes (Binney et al 1993, Barenblatt 1996) The general idea of renormalization groups is to simplify mathematically complex models that contain much fine-scale detail into simpler models and to develop scaling laws using similarity principles and techniques The simpler models (or equations) consist only of essential information of the phenomenon under study, and are able to describe and predict coarse-scale patterns with explicit scaling relations Renormalization group methods represent a fundamental concept and powerful technique in theoretical physics (Barenblatt 1996), which “make rigorous the scaling process through the derivation

of equations for blocks of cells in terms of the units that make them up” (Levin and Pacala 1997) Critical phenomena and phase transitions are common in ecology, particularly with spatial problems (Gardner et al 1987, Milne 1998), but only until

similarity of fluxes in terms of Reynolds number (Barenblatt 1996) The development

of Monin-Obukhov theory follows the general procedures of similarity analysis: (1) identifying the atmospheric processes that conform the dynamic similarity principle, (2) characterizing these processes with dimensionless similarity parameters (e.g., Reynolds number), (3) determining a set of scaling parameters (e.g., scaling wind velocity, scaling temperature, scaling humidity) and non-dimensionalized dependent and independent variables, and (4) deriving a set of similarity laws that are valid over a broad range of scales (Barenblatt 1996) By so doing, Monin-Obukhov theory relates turbulent fluxes in the surface layer to mean vertical gradients of wind, temperature, and specific humidity (Wu 1990)

recently have renormalization group methods been applied in ecological studies

(e.g., Levin and Pacala 1997, Milne 1998)

Gunther (1975) pointed out that “Structures and functions of all living beings,

irrespective of their size, can be studied by means of some basic physical methods,

viz., dimensional analysis and theories of similarity.” Although it is unlikely that all

structures and functions of the biological world (even at the organism level) can be

effectively studied by using dimensional analysis and similarity analysis alone, there

is little doubt that they will continue to play an important role in biological and

ecological scaling A great number of allometric studies in biology and ecology have

further demonstrated the power and elegance of similarity-based methods However,

the applicability and accuracy of these methods may depend on the levels of

biological organization and the variability of processes with scale In the following,

we turn our attention to some of the major issues in allometric scaling

2.2.3 Biological Allometry

Gould (1966) defined allometry as “the study of size and its consequences.”

Similarly, Niklas (1994) described allometry as “the study of size-correlated

variations in organic form and process.” Among other definitions of allometry is any

“departure from geometric similarity” (LaBarbera 1989) For several decades

allometry has focused primarily on the body size (or mass) of organisms as the

fundamental variable (Calder 1983, Peters 1983, Schmidt-Nielsen 1984) Niklas

(1994) summarized three meanings of allometry: (1) a relationship between the

growth of a part of an organism and the growth of the whole organism, (2) a

relationship between organism size and biological form and process, and (3) a

size-correlated relationship deviating from geometric similarity that is exhibited by

objects of varying sizes with the same geometry and shape Brown et al (2000)

noted that allometric studies in biology have been carried out at three levels of

biological organizations: within individual organisms (e.g., animal circulatory

networks and tree branching architecture), among individual organisms of different

sizes (e.g., body-size related variations in biological pattern and process), and within

populations or communities (e.g., allometric scaling of population density and

community biomass)

Allometric scaling is rooted in the concepts of similarity and, as in physical

sciences, allometric relations in biology usually take the form of a power law:

where Y is some biological variable, Y 0 is a normalization (or scaling) constant, M is

some size-related variable (usually body mass), and b is the scaling exponent

In Equation 2.1, if b = 1, the relationship becomes linear, and is called isometric

scaling; if b ≠ 1, then the relationship is either geometric scaling or allometric

scaling (including fractal scaling) Geometric (or Euclidean) scaling is based on complete similarity, whereas allometric scaling is based on incomplete similarity or

self-similarity (Barenblatt 1996, Schneider 2001a) For example, based on the geometric similarity of Euclidean objects we can analytically derive the following relationships among volume (V ), area (A), the length dimension (l ), and mass (M):

A ∝ l2, V ∝ l3, M ∝ V, l ∝ M1 / 3, and A ∝ M2 /3 These simple geometric scaling rules mean that, if objects of different sizes are completely similar, their liner dimensions and surface areas should be proportional to the 1/3 and 2/3 powers of their mass (assuming a constant density) In other words, if b = 1/3, Equation 2.1

suggests that a property of an object (Y ) is dependent on the length dimension of the

object (M ); if b = 2/3, then Equation 2.1 suggests that Y is dependent on the surface

area of the object However, Brown et al (2000, 2002), among others, argued that organisms do not seem to follow such simple geometric scaling rules; rather, they commonly exhibit “quarter-power scaling” relationships – i.e., the scaling exponent takes the value of simple multiples of 1/4 For example, b = 3/4 for the whole-

organism metabolic rates of a variety of animals ranging from mice to elephants; b = 1/4 for the heart rates of animals; b = −1/4 for the life span of animal species; b =

3/8 for the radius of the aorta of animals and the trunks of trees; and b = −3/4 for the

population density of animals (Brown et al 2000, Schmid et al 2000, Carbone and Gittleman 2002) While these scaling relations are general, variability can be substantial even for the same biological process For instance, LaBarbera (1989) reported that, for scaling of home range area with body size of terrestrial mammals,

b = 1.18 for herbivores, b = 1.51 for carnivores, b = 0.97 for omnivores, and b =

0.74, 1.39, or 1.65 for all mammals depending on data sets used for calculation One of the best-known examples of allometric scaling in plant ecology is the self-thinning law in plants In even-aged plant communities, the average biomass of individual plants (W) scales with plant density (D) following a power law:

biomass density This means that plant population density scales with plant weight with a scaling exponent of −2/3 (i.e., D ∝ W−2 / 3) rather than −3/4 as in animals This scaling relation was obtained from regression analysis based on empirical data

as well as analytical studies based on geometric similarity – the so-called “surface area law” (S ∝ V2 / 3, where V is the volume and S is the surface area; Niklas 1994)

While this biomass-density relation has been held as a “law” for decades, recent studies have found little empirical evidence to support its universality and consistency (Weller 1987, Zeide 1987, Lonsdale 1990) In particular, the scaling exponent is not a constant, but rather a variable that is influenced by the shade tolerance of plants under study and taxonomic groups of choice Zeide (1987) concluded that “the law is neither precise nor accurate,” and Lonsdale (1990) stated that, in the log-log plot of stand biomass vs plant density, “straight lines are the exception rather than the rule.”

Enquist et al (1998) showed that whole-plant resource use scales as W3 / 4 and that, accordingly, the scaling exponent for the biomass-density relation or the self-thinning law is −3/4 (i.e., D ∝ W−3 / 4), not −2/3 as previously reported Thus, they

concluded that plants do not differ from animals in terms of scaling of population

density with respect to body mass, confirming the prediction of their general

mechanistic model of resource use in fractal-like branching networks (West et al

1997) This model, however, has met an increasing number of criticisms claiming

2.2.4 Spatial Allometry

While sharing common features of similarity-based scaling methods, biological

allometry has focused primarily on body size Most of the allometric equations do

not directly address the problem of spatial scaling However, allometry as a general

method can be applied to spatial scaling when the independent variable is spatial

scale instead of body mass Such studies have been termed spatial allometry

(Schneider 2001a, b) In this case, the similarity principles pertain to the spatially

extended systems (e.g., habitats, landscapes) rather than the individual organisms A

general spatial allometric scaling relation can be written as follows:

S 0 and S, respectively, and β is the scaling exponent

In Equations 2.3 and 2.4, S and S 0 may be expressed as extent or grain size If S

is extent and S 0 is grain size, then the ratio, S/S 0, defines the spatial (or temporal)

scope (sensu Schneider 2001a), which is useful for comparing scaling studies among

different systems As with Equation 2.1, Equation 2.3 indicates isometric scaling

when β = 1, and geometric (Euclidean) or fractal scaling when β ≠ 1 Schneider

(2001b) pointed out that geometric scaling results when β is “an integer or ratio of

integers,” whereas fractal scaling is indicated by a value of β that is “not an

integer.” In practice, however, it is not a trivial matter to distinguish between a

“ratio of integers” and a “fractal” dimension Thus, inferring the nature of similarity

based merely on regression results, as often done in biological allometry, is not

warranted

Some allometric relations at the levels of populations and communities may be

directly related to spatial scaling For example, if population density scales with

body mass as D = D0M−0.75, one can derive a scaling relation between the total

number of animals (N) and habitat area (A): N = D0AM−0.75 or between the total

biomass (B) of the animal species and habitat area: B = DAM = D0AM0.25 If home

that it is mathematically flawed and empirically unwarranted (e.g., Magnani 1999,

Bokma 2004, Cyr and Walker 2004, Kozlowski and Konarzewski 2004) Nevertheless,

allometric scaling, as a general approach, remains useful, and its rule in spatial

scaling is discussed below

range scales with body mass as H ∝ M b, then population density can be directly related to the size of home range: D ∝ HM −(0.75+ b) The best known example of

spatial allometry, however, may well be the species-area relationship (SAR) SAR

is commonly described by a power-law function: z

S = cA , where c is a constant influenced by the effect of geographical variations on S, and z is the scaling

exponent with a value close to 0.25 SAR has been regarded as “ecology’s most general, yet protean pattern” (Lomolino 2000) and one of the few widely accepted laws in ecology (Schoener et al 2001)

Some recent studies suggested that SAR is an example of scale invariance that reflects self-similarity in species abundance and distribution (e.g., Harte et al 1999, Kunin 1999) However, many others have indicated that the value of the scaling exponent of SAR may vary widely and that the power-law scaling only holds over a finite range of spatial scales in real landscapes (Crawley and Harral 2001, Schoener

et al 2001) While scale invariant pattern is often believed to imply a single underlying process, SAR may have multiple scaling domains if examined over many orders of magnitude in space This observation favors the explanation that different factors determine species diversity at different ranges of scales (Shmida and Wilson

1985, Crawley and Harral 2001, Whittaker et al 2001) For example, Lomolino (2000) argued that, for isolated ecosystems, SAR has three fundamentally different realms: (1) erratic changes influenced by idiosyncratic differences among islands and random catastrophic disturbance events for small islands, (2) a monotonic deterministic pattern determined by island area and associated ecological factors for intermediate-sized islands, and (3) again a monotonically increasing pattern for

islands large enough for in situ speciation Nevertheless, as with the self-thinning

law, the debate and controversies on the universality, scale invariance, and ecological interpretation of SAR do not necessarily invalidate the use of the allometric scaling approach; it actually demonstrates its usefulness as a research tool

In landscape geomorphology, it has long been noted that landform attributes exhibit allometric relationships (Woldenberg 1969, Bull 1975, Church and Mark 1980) For example, Hood (2002) identified several allometric scaling relations between slough attributes (e.g., area, outlet width, perimeter, length) for rivers in the Pacific Northwest of the United States, and showed that detrital insect flotsam density was also allometrically related to slough perimeter In a recent study of the landscape dynamics of over 640 peatland bog pools in northern Scotland, Belyea and Lancaster (2002) found that the pools became deeper and more convoluted in shape with increasing size, and that the relationships between the area, depth, width, and length of the bog pools showed allometric (rather than geometric) scaling Schneider (2001a,b) provided a number of examples of spatial allometry for lake ecosystems and aquatic mesocosms in terms of the geometric attributes

of the systems (e.g., volume, area, perimeter, and depth of lakes or mesocosms) and biological properties (e.g., fish catch, primary production) In recent decades,

the allometric study of landform, or landscape allometry, has been elevated

to a new level of enthusiasm and insight by applying the concepts of fractals and

self-organization (Mandelbrot 1982, Turcotte 1995, Rodriguez-Iturbe and Rinaldo

1997, Phillips 1999, Schneider 2001a, b)

In landscape ecology, there have been many examples of spatial patterns

exhibiting allometric or fractal scaling relations (e.g., Milne 1991, Nikora et al

1999, Wu 2004) Although some authors attempt to associate power scaling

relations to underlying “universal” laws or scale invariance theories, such scaling

relations usually only hold for limited ranges of scale (Milne 1991, Berntson and

Stoll 1997, Wu 2004) Without resorting to any such grandiose assumptions,

however, spatial allometry can still be used as a valuable empirical scaling method

to summarize and extrapolate observed patterns over a range of scales, and to

provide clues about the underlying processes, using a “scalogram approach” (Wu

2004)

2.3 DYNAMIC MODEL-BASED SCALING METHODS

2.3.1 Some Concepts of Scaling with Dynamic Models

In contrast with similarity-based scaling methods that deal with complex phenomena

in a relatively simple manner, dynamic model-based scaling methods focus more on

the processes and mechanisms of the phenomena under study They may

incorporate, but do not rely on, similarity concepts in theory and dimensional

techniques in practice Dynamic models are composed of state variables, rate

variables, input variables, output variables, parameters, and constants Parameters

and constants help define rate variables and relate input and output variables to state

variables Because these terms are defined differently in the literature, some

clarifications are needed here to avoid confusion Following Bierkens et al (2000),

parameters may change in space, but not in time; constants are the only part of a

model that does not change in space and time (i.e., scale-invariant); and all other

model components may change in both space and time Dynamic models can be

implemented in mathematically explicit forms (e.g., differential or difference

equations) or mathematically implicit forms (e.g., mathematical relation-based or

rule-based simulation algorithms written in computer languages)

To illustrate different scaling methods clearly and precisely, let’s assume that a

dynamic model at a local scale s1 is:

1

where y( s1), v, θ, and i are the output variables, state variables, parameters, and

input variables at scale s1, respectively

Also, let’s assume that a model can be developed at a broader scale s2 as:

2